Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2166,2,Mod(1,2166)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2166.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2166, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2166.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,-3,-3,3,0,3,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(17.2955970778\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.87939\) of defining polynomial
Character \(\chi\) \(=\) 2166.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.41147 q^{5} +1.00000 q^{6} -2.87939 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.41147 q^{10} -0.347296 q^{11} -1.00000 q^{12} -1.65270 q^{13} +2.87939 q^{14} +3.41147 q^{15} +1.00000 q^{16} +6.94356 q^{17} -1.00000 q^{18} -3.41147 q^{20} +2.87939 q^{21} +0.347296 q^{22} +6.80066 q^{23} +1.00000 q^{24} +6.63816 q^{25} +1.65270 q^{26} -1.00000 q^{27} -2.87939 q^{28} +6.35504 q^{29} -3.41147 q^{30} +1.59627 q^{31} -1.00000 q^{32} +0.347296 q^{33} -6.94356 q^{34} +9.82295 q^{35} +1.00000 q^{36} -11.2121 q^{37} +1.65270 q^{39} +3.41147 q^{40} -3.49020 q^{41} -2.87939 q^{42} +2.28312 q^{43} -0.347296 q^{44} -3.41147 q^{45} -6.80066 q^{46} +5.59627 q^{47} -1.00000 q^{48} +1.29086 q^{49} -6.63816 q^{50} -6.94356 q^{51} -1.65270 q^{52} +1.98040 q^{53} +1.00000 q^{54} +1.18479 q^{55} +2.87939 q^{56} -6.35504 q^{58} +0.445622 q^{59} +3.41147 q^{60} -12.5321 q^{61} -1.59627 q^{62} -2.87939 q^{63} +1.00000 q^{64} +5.63816 q^{65} -0.347296 q^{66} -1.07873 q^{67} +6.94356 q^{68} -6.80066 q^{69} -9.82295 q^{70} -16.6236 q^{71} -1.00000 q^{72} +12.4192 q^{73} +11.2121 q^{74} -6.63816 q^{75} +1.00000 q^{77} -1.65270 q^{78} -10.9240 q^{79} -3.41147 q^{80} +1.00000 q^{81} +3.49020 q^{82} +11.7169 q^{83} +2.87939 q^{84} -23.6878 q^{85} -2.28312 q^{86} -6.35504 q^{87} +0.347296 q^{88} +1.79292 q^{89} +3.41147 q^{90} +4.75877 q^{91} +6.80066 q^{92} -1.59627 q^{93} -5.59627 q^{94} +1.00000 q^{96} +3.65270 q^{97} -1.29086 q^{98} -0.347296 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{6} - 3 q^{7} - 3 q^{8} + 3 q^{9} - 3 q^{12} - 6 q^{13} + 3 q^{14} + 3 q^{16} + 6 q^{17} - 3 q^{18} + 3 q^{21} + 6 q^{23} + 3 q^{24} + 3 q^{25} + 6 q^{26} - 3 q^{27}+ \cdots + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −3.41147 −1.52566 −0.762829 − 0.646601i \(-0.776190\pi\)
−0.762829 + 0.646601i \(0.776190\pi\)
\(6\) 1.00000 0.408248
\(7\) −2.87939 −1.08831 −0.544153 − 0.838986i \(-0.683149\pi\)
−0.544153 + 0.838986i \(0.683149\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 3.41147 1.07880
\(11\) −0.347296 −0.104714 −0.0523569 − 0.998628i \(-0.516673\pi\)
−0.0523569 + 0.998628i \(0.516673\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.65270 −0.458378 −0.229189 − 0.973382i \(-0.573607\pi\)
−0.229189 + 0.973382i \(0.573607\pi\)
\(14\) 2.87939 0.769548
\(15\) 3.41147 0.880839
\(16\) 1.00000 0.250000
\(17\) 6.94356 1.68406 0.842031 − 0.539430i \(-0.181360\pi\)
0.842031 + 0.539430i \(0.181360\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0 0
\(20\) −3.41147 −0.762829
\(21\) 2.87939 0.628333
\(22\) 0.347296 0.0740438
\(23\) 6.80066 1.41804 0.709018 − 0.705191i \(-0.249139\pi\)
0.709018 + 0.705191i \(0.249139\pi\)
\(24\) 1.00000 0.204124
\(25\) 6.63816 1.32763
\(26\) 1.65270 0.324122
\(27\) −1.00000 −0.192450
\(28\) −2.87939 −0.544153
\(29\) 6.35504 1.18010 0.590050 − 0.807366i \(-0.299108\pi\)
0.590050 + 0.807366i \(0.299108\pi\)
\(30\) −3.41147 −0.622847
\(31\) 1.59627 0.286698 0.143349 − 0.989672i \(-0.454213\pi\)
0.143349 + 0.989672i \(0.454213\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0.347296 0.0604565
\(34\) −6.94356 −1.19081
\(35\) 9.82295 1.66038
\(36\) 1.00000 0.166667
\(37\) −11.2121 −1.84326 −0.921632 − 0.388066i \(-0.873143\pi\)
−0.921632 + 0.388066i \(0.873143\pi\)
\(38\) 0 0
\(39\) 1.65270 0.264644
\(40\) 3.41147 0.539401
\(41\) −3.49020 −0.545078 −0.272539 − 0.962145i \(-0.587863\pi\)
−0.272539 + 0.962145i \(0.587863\pi\)
\(42\) −2.87939 −0.444299
\(43\) 2.28312 0.348172 0.174086 − 0.984730i \(-0.444303\pi\)
0.174086 + 0.984730i \(0.444303\pi\)
\(44\) −0.347296 −0.0523569
\(45\) −3.41147 −0.508553
\(46\) −6.80066 −1.00270
\(47\) 5.59627 0.816299 0.408150 − 0.912915i \(-0.366174\pi\)
0.408150 + 0.912915i \(0.366174\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.29086 0.184408
\(50\) −6.63816 −0.938777
\(51\) −6.94356 −0.972293
\(52\) −1.65270 −0.229189
\(53\) 1.98040 0.272029 0.136014 − 0.990707i \(-0.456571\pi\)
0.136014 + 0.990707i \(0.456571\pi\)
\(54\) 1.00000 0.136083
\(55\) 1.18479 0.159757
\(56\) 2.87939 0.384774
\(57\) 0 0
\(58\) −6.35504 −0.834457
\(59\) 0.445622 0.0580151 0.0290075 − 0.999579i \(-0.490765\pi\)
0.0290075 + 0.999579i \(0.490765\pi\)
\(60\) 3.41147 0.440419
\(61\) −12.5321 −1.60457 −0.802285 − 0.596941i \(-0.796382\pi\)
−0.802285 + 0.596941i \(0.796382\pi\)
\(62\) −1.59627 −0.202726
\(63\) −2.87939 −0.362768
\(64\) 1.00000 0.125000
\(65\) 5.63816 0.699327
\(66\) −0.347296 −0.0427492
\(67\) −1.07873 −0.131787 −0.0658937 − 0.997827i \(-0.520990\pi\)
−0.0658937 + 0.997827i \(0.520990\pi\)
\(68\) 6.94356 0.842031
\(69\) −6.80066 −0.818703
\(70\) −9.82295 −1.17407
\(71\) −16.6236 −1.97286 −0.986430 − 0.164185i \(-0.947501\pi\)
−0.986430 + 0.164185i \(0.947501\pi\)
\(72\) −1.00000 −0.117851
\(73\) 12.4192 1.45356 0.726780 − 0.686871i \(-0.241016\pi\)
0.726780 + 0.686871i \(0.241016\pi\)
\(74\) 11.2121 1.30338
\(75\) −6.63816 −0.766508
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) −1.65270 −0.187132
\(79\) −10.9240 −1.22904 −0.614521 − 0.788901i \(-0.710651\pi\)
−0.614521 + 0.788901i \(0.710651\pi\)
\(80\) −3.41147 −0.381414
\(81\) 1.00000 0.111111
\(82\) 3.49020 0.385428
\(83\) 11.7169 1.28609 0.643047 − 0.765826i \(-0.277670\pi\)
0.643047 + 0.765826i \(0.277670\pi\)
\(84\) 2.87939 0.314167
\(85\) −23.6878 −2.56930
\(86\) −2.28312 −0.246195
\(87\) −6.35504 −0.681331
\(88\) 0.347296 0.0370219
\(89\) 1.79292 0.190049 0.0950245 − 0.995475i \(-0.469707\pi\)
0.0950245 + 0.995475i \(0.469707\pi\)
\(90\) 3.41147 0.359601
\(91\) 4.75877 0.498855
\(92\) 6.80066 0.709018
\(93\) −1.59627 −0.165525
\(94\) −5.59627 −0.577211
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 3.65270 0.370876 0.185438 − 0.982656i \(-0.440630\pi\)
0.185438 + 0.982656i \(0.440630\pi\)
\(98\) −1.29086 −0.130396
\(99\) −0.347296 −0.0349046
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2166.2.a.o.1.1 3
3.2 odd 2 6498.2.a.bs.1.3 3
19.6 even 9 114.2.i.d.55.1 ✓ 6
19.16 even 9 114.2.i.d.85.1 yes 6
19.18 odd 2 2166.2.a.u.1.1 3
57.35 odd 18 342.2.u.a.199.1 6
57.44 odd 18 342.2.u.a.55.1 6
57.56 even 2 6498.2.a.bn.1.3 3
76.35 odd 18 912.2.bo.f.769.1 6
76.63 odd 18 912.2.bo.f.625.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.i.d.55.1 ✓ 6 19.6 even 9
114.2.i.d.85.1 yes 6 19.16 even 9
342.2.u.a.55.1 6 57.44 odd 18
342.2.u.a.199.1 6 57.35 odd 18
912.2.bo.f.625.1 6 76.63 odd 18
912.2.bo.f.769.1 6 76.35 odd 18
2166.2.a.o.1.1 3 1.1 even 1 trivial
2166.2.a.u.1.1 3 19.18 odd 2
6498.2.a.bn.1.3 3 57.56 even 2
6498.2.a.bs.1.3 3 3.2 odd 2